{"id":38369,"date":"2013-08-16T13:16:45","date_gmt":"2013-08-16T10:16:45","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/mathinfo042013\/"},"modified":"2019-06-24T17:33:15","modified_gmt":"2019-06-24T14:33:15","slug":"mathinfo042013","status":"publish","type":"post","link":"https:\/\/vocedu.azbuki.bg\/en\/matematics\/matharticles\/mathinfo042013\/","title":{"rendered":"Mathematics and Informatics, Number 4\/2013, Volume 56"},"content":{"rendered":"<p><strong><a name=\"top\">&nbsp;<\/a>\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS<\/strong><br \/><a href=\"#art01\">\u041f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0430 \u043a \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u043e\u043c \u2013 \u0443\u0434\u043e\u0431\u043d\u044b\u0439 \u043f\u043e\u0432\u043e\u0434 \u043e\u0431\u0441\u0443\u0434\u0438\u0442\u044c \u0442\u043e\u043d\u043a\u043e\u0441\u0442\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u0439 <\/a><br \/><a href=\"#art01\">[Preparation for Olympiads \u2013 a Convenient Cause to Discuss&nbsp;Refinements of Mathematical Reasonings] \/ <\/a><br \/><em>\u0412\u044f\u0447\u0435\u0441\u043b\u0430\u0432 \u0422\u0435\u043f\u043b\u044f\u043a\u043e\u0432, Vjacheslav&nbsp;Tepljakov \/ \u041d\u0438\u043d\u0430 \u041f\u0430\u0442\u0440\u043e\u043d\u043e\u0432\u0430, Nina Patronova<\/em> &#8211; \u0441\u0442\u0440. 305<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/vocedu.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art02\">\u0412\u044a\u0440\u0445\u0443 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u0441\u044a\u0441 \u0437\u0440\u0438\u0442\u0435\u043b\u043d\u0438 \u0443\u0432\u0440\u0435\u0436\u0434\u0430\u043d\u044f<\/a><br \/><a href=\"#art02\">[On Mathematics Teaching to Students with Visual Impairments] \/ <\/a><br \/><em>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437<\/em><em>\u0434\u0435\u0432, Sava Grozdev\/ \u0415\u043b\u0438\u0437\u0430 \u041f\u0435\u0442\u0440\u043e\u0432\u0430, Eliza Petrova<\/em> &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;314<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/vocedu.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art03\">\u0417\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043d\u0430 \u0440\u043e\u043c\u0438 (5. \u2013 7. \u041a\u043b\u0430\u0441) <\/a><br \/><a href=\"#art03\">[On Mathematics Education&nbsp;for Gipsy Students (5th \u2013 7th Grades)] \/ <\/a><br \/><em>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, Sava Grozdev\/\u0414\u0438\u0430\u043d\u0430&nbsp;\u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, Diana Stefanova<\/em> &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;325<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/vocedu.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art04\">\u0420\u0430\u0437\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 \u0438 \u043f\u0440\u0430\u0432\u0438 \u0432 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430 <\/a><br \/><a href=\"#art04\">[Location of&nbsp;Remarkable Points and Straight Lines in the Triangle] \/ <\/a><br \/><em>\u0425\u0440\u0438\u0441\u0442\u043e \u041b\u0435\u0441\u043e\u0432, Hristo&nbsp;Lessov<\/em> &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;338<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/vocedu.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art05\">\u0421\u0432\u0435\u0436\u0434\u0430\u043d\u0435 \u043d\u0430 \u0438\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0438 \u0442\u0440\u0430\u043d\u0441\u0446\u0435\u043d\u0434\u0435\u043d\u0442\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0438 \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u0434\u043e&nbsp;\u043c\u043e\u0434\u0443\u043b\u043d\u0438<\/a><br \/> <a href=\"#art05\">[Reduction of Irrational and Transcendental Equations and Inequalities&nbsp;to Absolute Value Ones] \/<\/a><br \/> <em>\u041f\u0435\u043d\u043a\u0430 \u0420\u0430\u043d\u0433\u0435\u043b\u043e\u0432\u0430, Penka Rangelova<\/em> &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;343<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/vocedu.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art06\">\u0415\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0432 \u0441\u0440\u0435\u0434\u043d\u043e\u0442\u043e \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u0441 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0438 \u0442\u0430\u0431\u043b\u0438\u0446\u0438 <\/a><br \/><a href=\"#art06\">[Extremal Problems in High School by Computer Tables] \/ <\/a><br \/><em>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432,&nbsp;Sava Grozdev \/ \u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432, Deko Dekov<\/em> &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;351<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/vocedu.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art08\">\u041f\u044a\u0440\u0432\u0438\u044f\u0442 \u0431\u044a\u043b\u0433\u0430\u0440\u0441\u043a\u0438 \u0443\u0447\u0435\u0431\u043d\u0438\u043a \u043f\u043e \u0442\u0435\u043e\u0440\u0438\u044f \u043d\u0430 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438\u0442\u0435 <\/a><br \/><a href=\"#art08\">[The First Bulgarian&nbsp;Textbook on Probability Theory] \/ <\/a><br \/><em>\u041f\u043b\u0430\u043c\u0435\u043d \u041c\u0430\u0442\u0435\u0435\u0432, Plamen Mateev<\/em> &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;384<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/vocedu.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><strong>\u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418 [CONTEST PROBLEMS]<\/strong><br \/><em>\u0420\u0443\u0431\u0440\u0438\u043a\u0430\u0442\u0430 \u0441\u0435 \u0432\u043e\u0434\u0438 \u043e\u0442 \u0434-\u0440 \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \u0438 \u0434-\u0440 \u0416\u0438\u0432\u043a\u043e \u0416\u0435\u043b\u0435\u0432<\/em><\/p>\n<p>\u041a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u043d\u0430 \u0431\u0440\u043e\u044f [Contest problems of the issue] &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;392<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 \u0431\u0440\u043e\u0439 5, 2013 [Solutions of the contest&nbsp;problems from issue 5, 2013] &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;393<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>\u0412 \u041d\u041e\u0412\u0418\u0422\u0415 \u0411\u0420\u041e\u0415\u0412\u0415 \u041d\u0410 \u0421\u041f\u0418\u0421\u0410\u041d\u0418\u042f\u0422\u0410 \u041d\u0410\u0418\u0417\u0414\u0410 \u0422\u0415\u041b\u0421\u0422\u0412\u041e \u201e\u0410\u0417 \u0411\u0423\u041a\u0418 \u201d \u0427\u0415\u0422\u0415\u0422\u0415 \/<\/strong><br \/><strong> READ IN THE LATEST ISSUES OF \u201eAZ BUKI\u201d JOURNALS<\/strong> &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;395<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>\u0423\u041a\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415 \/ GUIDE FOR AUTHORS<\/strong> &#8211;&nbsp;\u0441\u0442\u0440.&nbsp;397<\/p>\n<p>&nbsp;<\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong><a name=\"art01\">&nbsp;<\/a>\u041f\u041e\u0414\u0413\u041e\u0422\u041e\u0412\u041a\u0410 \u041a \u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u0410\u041c&nbsp;\u0423\u0414\u041e\u0411\u041d\u042b\u0419 \u041f\u041e\u0412\u041e\u0414 \u041e\u0411\u0421\u0423\u0414\u0418\u0422\u042c \u0422\u041e\u041d\u041a\u041e\u0421\u0422\u0418&nbsp;\u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u0427\u0415\u0421\u041a\u0418\u0425 \u0420\u0410\u0421\u0421\u0423\u0416\u0414\u0415\u041d\u0418\u0419<\/strong><\/p>\n<p><em>\u0412\u044f\u0447\u0435\u0441\u043b\u0430\u0432 \u0422\u0435\u043f\u043b\u044f\u043a\u043e\u0432, \u041d\u0438\u043d\u0430 \u041f\u0430\u0442\u0440\u043e\u043d\u043e\u0432\u0430<\/em><br \/><em>\u201e\u0414\u0430\u0436\u0435 \u043d\u0435 \u043f\u043b\u0430\u0432\u0430\u044f \u043d\u0430 \u0433\u043b\u0443\u0431\u043e\u043a\u0438\u0445<\/em><br \/><em>\u043c\u0435\u0441\u0442\u0430\u0445, \u043f\u043e\u043b\u0435\u0437\u043d\u043e \u0437\u043d\u0430\u0442\u044c, \u0433\u0434\u0435 \u043e\u043d\u0438\u201c<\/em><br \/><strong>\u041a\u043e\u0437\u044c\u043c\u0430 \u041f\u0440\u0443\u0442\u043a\u043e\u0432 (\u201e\u041c\u044b\u0441\u043b\u0438 \u0438 \u0430\u0444\u043e\u0440\u0438\u0437\u043c\u044b\u201c)<\/strong><\/p>\n<p><strong>Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u044c\u0435 \u0440\u0430\u0441\u0441\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043e \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0437\u0430\u0434\u0430\u0447\u0430\u0445, \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u0432\u0448\u0438\u0445\u0441\u044f \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u043c \u043a\u0440\u0443\u0436\u043a\u0435 \u0434\u043b\u044f \u0448\u043a\u043e\u043b\u044c\u043d\u0438\u043a\u043e\u0432 \u0432 \u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442\u0435 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438, \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0445&nbsp;\u0438 \u043a\u043e\u0441\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0439 \u0421\u0435\u0432\u0435\u0440\u043d\u043e\u0433\u043e (\u0410\u0440\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e) \u0444\u0435\u0434\u0435\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442\u0430&nbsp;\u0438\u043c\u0435\u043d\u0438 \u041c. \u0412. \u041b\u043e\u043c\u043e\u043d\u043e\u0441\u043e\u0432\u0430.<\/p>\n<p><em>Keywords<\/em>: proof, existence, Euler characteristic, Vieta\u2019s formulae, isoperimetric&nbsp;problem<\/p>\n<p>&nbsp;<\/p>\n<p><strong>PREPARATION FOR OLYMPIADS \u2013 A CONVENIENT CAUSE&nbsp;TO DISCUSS REFINEMENTS OF MATHEMATICAL REASONINGS<\/strong><\/p>\n<p><strong>Vjacheslav Tepljakov, Nina Patronova<br \/><\/strong><br \/><strong>Abstract.<\/strong> The paper considers some mathematical problems which have been&nbsp;discussed at mathematical circles for high school students in the Institute of Mathematics,&nbsp;Information and Space Technologies of the Northern (Arctic) Federal University named&nbsp;after M. V. Lomonosov.<\/p>\n<p><strong>Vjacheslav Tepljakov<\/strong><br \/>\u0002Associated Professor<br \/>Department of Algebra and Geometry<br \/>Institute of Mathematics and Computer Science<\/p>\n<p><strong>Nina Patronova<\/strong><br \/>Associated Professor, PhD in Pedagogy<br \/>Department of Methodics of Mathematics Teaching<br \/>Institute of Mathematics and Computer Science<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong><a name=\"art02\">&nbsp;<\/a>\u0412\u042a\u0420\u0425\u0423 \u041f\u0420\u0415\u041f\u041e\u0414\u0410\u0412\u0410\u041d\u0415\u0422\u041e \u041d\u0410 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410&nbsp;<\/strong><strong>\u041d\u0410 \u0423\u0427\u0415\u041d\u0418\u0426\u0418 \u0421\u042a\u0421 \u0417\u0420\u0418\u0422\u0415\u041b\u041d\u0418 \u0423\u0412\u0420\u0415\u0416\u0414\u0410\u041d\u0418\u042f<br \/><\/strong><br \/><strong>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0415\u043b\u0438\u0437\u0430 \u041f\u0435\u0442\u0440\u043e\u0432\u0430<br \/><\/strong><br \/><strong>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430, \u043f\u0440\u0435\u0434\u043d\u0430\u0437\u043d\u0430\u0447\u0435\u043d\u0438 \u0437\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u0441\u044a\u0441 \u0437\u0440\u0438\u0442\u0435\u043b\u043d\u0438 \u0443\u0432\u0440\u0435\u0436\u0434\u0430\u043d\u0438\u044f. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0430 \u0435 \u0441\u0445\u0435\u043c\u0430 \u0437\u0430 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435&nbsp;\u043d\u0430 \u0443\u0440\u043e\u043a\u0430 \u201e\u041f\u0438\u0442\u0430\u0433\u043e\u0440\u043e\u0432\u0430 \u0442\u0435\u043e\u0440\u0435\u043c\u0430\u201c. \u0421\u044a\u043e\u0442\u0432\u0435\u0442\u043d\u0430\u0442\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430 \u0441\u0435 \u043e\u0441\u043d\u043e\u0432\u0430\u0432\u0430 \u043d\u0430 \u043f\u0440\u0430\u0432\u043e\u044a\u0433\u044a\u043b&nbsp;\u043d\u0438\u044f \u043c\u0435\u0442\u043e\u0434.<\/p>\n<p><em>Keywords<\/em>: students with visual impairments, mathematics, Pythagorean theorem,&nbsp;rectangular method<\/p>\n<p>&nbsp;<\/p>\n<p><strong>ON MATHEMATICS TEACHING&nbsp;<\/strong><strong>TO STUDENTS WITH VISUAL IMPAIRMENTS<br \/><\/strong><br \/><strong>Abstract.<\/strong> The paper considers several exercises in Mathematics for students with&nbsp;visual impairments. A scheme is proposed to teach the lesson \u201eThe Pythagorean theorem\u201c.&nbsp;The corresponding methodology is based on the rectangular method.<\/p>\n<p><strong>Sava Grozde<\/strong>v<br \/>Professor, Doctor in Mathematics, DSc in Pedagogy<br \/>Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p><strong>Eliza Petrova<\/strong><br \/>teacher<br \/>\u201eLouis Braille\u201c Secondary School<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong><a name=\"art03\">&nbsp;<\/a>\u0417\u0410 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415\u0422\u041e \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410&nbsp;\u041d\u0410 \u0420\u041e\u041c\u0418 (5. \u2013 7. \u041a\u041b\u0410\u0421)<br \/><\/strong><br \/><strong>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430<br \/><\/strong><br \/><strong>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0430 \u0437\u0430 \u043f\u043e\u0432\u0438\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u0442\u0430&nbsp;\u043a\u0443\u043b\u0442\u0443\u0440\u0430 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043e\u0442 \u0440\u043e\u043c\u0441\u043a\u0438 \u043f\u0440\u043e\u0438\u0437\u0445\u043e\u0434 \u0447\u0440\u0435\u0437 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u043f\u043e\u0434\u0445\u043e\u0434\u0438 \u043d\u0430&nbsp;\u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435. \u041a\u0430\u0442\u043e \u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u0438 \u043d\u0430 \u043a\u0443\u043b\u0442\u0443\u0440\u0430\u0442\u0430 \u0441\u0430 \u0432\u0437\u0435\u0442\u0438 \u043f\u0440\u0435\u0434\u0432\u0438\u0434 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0438 \u043d\u0430 \u043c\u0438\u0441\u043b\u0435\u043d\u0435\u0442\u043e&nbsp;\u0438 \u0440\u0435\u0447\u0442\u0430.<\/p>\n<p><em>Keywords<\/em>: students of gipsy origin, thinking, speech, mathematical culture, reformulation.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>ON MATHEMATICS EDUCATION FOR GIPSY STUDENTS&nbsp;(5th \u2013 7th GRADES )<br \/><\/strong><br \/><strong>Abstract.<\/strong> The paper is dedicated to the problem of Mathematics culture improvement&nbsp;of gipsy students by various methodological approaches of teaching. Elements of&nbsp;thinking and speech are considered as components of culture.<\/p>\n<p><strong>Sava Grozdev<\/strong><br \/>Professor, Doktor in Mathematics, DSc in Pedagogy<br \/>Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p><strong>Diana Stefanova<\/strong><br \/>Teacher<br \/>Primary School \u201eNikola Vaptsarov\u201c<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong><a name=\"art04\">&nbsp;<\/a>\u0420\u0410\u0417\u041f\u041e\u041b\u041e\u0416\u0415\u041d\u0418\u042f \u041d\u0410 \u0417\u0410\u0411\u0415\u041b\u0415\u0416\u0418\u0422\u0415\u041b\u041d\u0418&nbsp;<\/strong><strong>\u0422\u041e\u0427\u041a\u0418 \u0418 \u041f\u0420\u0410\u0412\u0418 \u0412 \u0422\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u041a\u0410<br \/><\/strong><br \/><strong>\u0425\u0440\u0438\u0441\u0442\u043e \u041b\u0435\u0441\u043e\u0432<br \/><\/strong><br \/><strong>Abstract.<\/strong> \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 ABC \u0438 \u043d\u0435\u0433\u043e\u0432\u0438\u0442\u0435 \u043e\u0441\u043d\u043e\u0432\u043d\u0438 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438-&nbsp;\u0442\u0435\u043b\u043d\u0438 \u0442\u043e\u0447\u043a\u0438: \u043c\u0435\u0434\u0438\u0446\u0435\u043d\u0442\u044a\u0440 G (\u043f\u0440\u0435\u0441\u0435\u0447\u043d\u0430\u0442\u0430 \u0442\u043e\u0447\u043a\u0430 \u043d\u0430 \u043c\u0435\u0434\u0438\u0430\u043d\u0438\u0442\u0435), \u043e\u0440\u0442\u043e\u0446\u0435\u043d\u0442\u044a\u0440 H (\u043f\u0440\u0435\u0441\u0435\u0447\u043d\u0430\u0442\u0430 \u0442\u043e\u0447\u043a\u0430 \u043d\u0430 \u0432\u0438\u0441\u043e\u0447\u0438\u043d\u0438\u0442\u0435), \u0446\u0435\u043d\u0442\u044a\u0440 O \u043d\u0430 \u043e\u043f\u0438\u0441\u0430\u043d\u0430\u0442\u0430 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442 (\u043f\u0440\u0435\u0441\u0435\u0447\u043d\u0430\u0442\u0430 \u0442\u043e\u0447\u043a\u0430 \u043d\u0430&nbsp;\u0441\u0438\u043c\u0435\u0442\u0440\u0430\u043b\u0438\u0442\u0435), \u0446\u0435\u043d\u0442\u044a\u0440 I \u043d\u0430 \u0432\u043f\u0438\u0441\u0430\u043d\u0430\u0442\u0430 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442 (\u043f\u0440\u0435\u0441\u0435\u0447\u043d\u0430 \u0442\u043e\u0447\u043a\u0430 \u043d\u0430 \u044a\u0433\u043b\u043e\u043f\u043e\u043b\u043e\u0432\u044f\u0449\u0438\u0442\u0435).&nbsp;\u0418\u0437\u0432\u0435\u0434\u0435\u043d\u0438 \u0441\u0430 \u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e\u0441\u0442\u0438, \u043a\u043e\u0438\u0442\u043e \u0441\u0430 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u0438 \u0438 \u0434\u043e\u0441\u0442\u0430\u0442\u044a\u0447\u043d\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f \u0437\u0430 \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u043d\u0438&nbsp;\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u043d\u0430 \u043f\u0440\u0430\u0432\u0438\u0442\u0435, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438 \u043e\u0442 \u0442\u0435\u0437\u0438 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0438 \u0442\u043e\u0447\u043a\u0438.&nbsp;<\/p>\n<p><em>Keywords<\/em>: triangle, remarkable points, straight lines, special locations<\/p>\n<p>&nbsp;<\/p>\n<p><strong>LOCATION OF REMARKABLE&nbsp;<\/strong><strong>POINTS AND STRAIGHT LINES IN THE TRIANGLE<br \/><\/strong><br \/><strong>Abstract.<\/strong> The present paper considers \uf044ABC and its main remarkable points: the&nbsp;gravity center G (the common point of the medians), the orthocenter H (the common point&nbsp;of the altitudes), the circumcenter O (the common point of the perpendicular bisectors),&nbsp;the in-center I (the common point of the angle bisectors). Relations are derived, which&nbsp;represent necessary and sufficient conditions for special locations of the straight lines&nbsp;through these remarkable points.<\/p>\n<p><strong>Hristo Lessov<\/strong><br \/>\u0002Teacher<br \/>Mathematical School \u201eN. Obreshkov\u201c<br \/>Kazanlak, Bulgaria<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong><a name=\"art05\">&nbsp;<\/a>\u0421\u0412\u0415\u0416\u0414\u0410\u041d\u0415 \u041d\u0410 \u0418\u0420\u0410\u0426\u0418\u041e\u041d\u0410\u041b\u041d\u0418&nbsp;\u0418 \u0422\u0420\u0410\u041d\u0421\u0426\u0415\u041d\u0414\u0415\u041d\u0422\u041d\u0418 \u0423\u0420\u0410\u0412\u041d\u0415\u041d\u0418\u042f&nbsp;\u0418 \u041d\u0415\u0420\u0410\u0412\u0415\u041d\u0421\u0422\u0412\u0410 \u0414\u041e \u041c\u041e\u0414\u0423\u041b\u041d\u0418<\/strong><\/p>\n<p><strong>\u041f\u0435\u043d\u043a\u0430 \u0420\u0430\u043d\u0433\u0435\u043b\u043e\u0432\u0430<br \/><\/strong><br \/><strong>Abstract.<\/strong> \u0421 \u0440\u0430\u0437\u043d\u043e\u043e\u0431\u0440\u0430\u0437\u043d\u0438 \u043f\u0440\u0438\u043c\u0435\u0440\u0438 \u0435 \u043f\u043e\u043a\u0430\u0437\u0430\u043d\u043e \u0441\u0432\u0435\u0436\u0434\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0438\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0438&nbsp;\u0442\u0440\u0430\u043d\u0441 \u0446\u0435\u043d\u0434\u0435\u043d\u0442 \u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0438 \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u0434\u043e \u043c\u043e\u0434\u0443\u043b\u043d\u0438. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0437\u0430&nbsp;\u0441\u0430\u043c\u043e \u0441\u0442\u043e\u044f \u0442\u0435\u043b\u043d\u0430 \u0440\u0430\u0431\u043e\u0442\u0430.<\/p>\n<p><em>Keywords<\/em>: absolute value, equation, inequality, irrational, transcendental<\/p>\n<p>&nbsp;<\/p>\n<p><strong>R\u0415DUCTION OF IRRATIONAL AND TRANSCENDENTAL&nbsp;<\/strong><strong>EQUATIONS <br \/>AND INEQUALITIES TO ABSOLUTE VALUE ONES<br \/><\/strong><br \/><strong>Abstract.<\/strong> Reduction of irrational and transcendental equations and inequalities to&nbsp;absolute value ones is considered by various examples. Some tasks for individual work&nbsp;are proposed too.<\/p>\n<p><strong>Penka Rangelova<\/strong><br \/>\u0002Associated Professor, PhD<br \/>Faculty of Mathematics and Informatics<br \/>Plovdiv University \u201ePaisii Hiledarski\u201c<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong><a name=\"art06\">&nbsp;<\/a>\u0415\u041a\u0421\u0422\u0420\u0415\u041c\u0410\u041b\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418 \u0412 \u0421\u0420\u0415\u0414\u041d\u041e\u0422\u041e \u0423\u0427\u0418\u041b\u0418\u0429\u0415&nbsp;<\/strong><strong>\u0421 \u041f\u041e\u041c\u041e\u0429\u0422\u0410 \u041d\u0410 \u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u041d\u0418 \u0422\u0410\u0411\u041b\u0418\u0426\u0418<br \/><\/strong><br \/><strong>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432<br \/><\/strong><br \/><strong>Abstract.<\/strong> \u0410\u0432\u0442\u043e\u0440\u0438\u0442\u0435 \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0442 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0435\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0432 \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u0434\u0430&nbsp;\u0437\u0430\u043f\u043e\u0447\u0432\u0430 \u043d\u0430 \u043f\u043e-\u0440\u0430\u043d\u0435\u043d \u0435\u0442\u0430\u043f, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 \u043e\u0442 IX \u043a\u043b\u0430\u0441. \u0417\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u0438\u043c \u043c\u043e\u0436\u0435 \u0434\u0430 \u0441\u0435 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430&nbsp;\u0447\u0438\u0441\u043b\u0435\u043d \u043c\u0435\u0442\u043e\u0434, \u043f\u0440\u0438 \u043a\u043e\u0439\u0442\u043e \u0441 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430 \u0441\u0435 \u0447\u0435\u0440\u0442\u0430\u044f\u0442 \u0433\u0440\u0430\u0444\u0438\u043a\u0438 \u043d\u0430&nbsp;\u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438 \u0441\u0435 \u0441\u044a\u0437\u0434\u0430\u0432\u0430\u0442 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0438 \u0442\u0430\u0431\u043b\u0438\u0446\u0438. \u041f\u0440\u0438 \u0442\u043e\u0432\u0430 \u043d\u0435 \u0435 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u043f\u0440\u0435\u0434\u0432\u0430\u0440\u0438\u0442\u0435\u043b\u043d\u043e \u043f\u043e\u0437\u043d\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430 \u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044f. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u0442\u043e\u0437\u0438 \u043f\u043e\u0434\u0445\u043e\u0434.<\/p>\n<p><em>\u041aeywords<\/em>: extremal problem, numerical method, computer table, computer&nbsp;mathematics, high school.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>EXTREMAL PROBLEMS IN HIGH SCHOOL&nbsp;<\/strong><strong>BY COMPUTER TABLES<br \/><\/strong><br \/><strong>Abstract.<\/strong> The authors propose to start the study of extremal problems in school&nbsp;earlier, for example from the 9th grade on. A numerical method could be used for the&nbsp;purpose, which allows application of computer programs for graphical representation of&nbsp;functions and creation of computer tables. Thus, preliminary knowledge of derivatives is&nbsp;not necessary. The paper is dedicated to such an approach.&nbsp;<\/p>\n<p><strong>Sava Grozdev<\/strong><br \/>\u0002Professor, Doctor in Mathematics, DSc in Pedagogy<br \/>Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p><strong>Deko Dekov<\/strong><br \/>Associated Professor, PhD in Mathematics<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong><a name=\"art07\">&nbsp;<\/a>\u0410\u0420\u0425\u0418\u0422\u0415\u041a\u0422\u0423\u0420\u0410 \u201e\u041c\u041e\u0414\u0415\u041b-\u0418\u0417\u0413\u041b\u0415\u0414-\u041a\u041e\u041d\u0422\u0420\u041e\u041b\u0415\u0420\u201c&nbsp;<\/strong><strong>\u0412 \u041f\u041e\u041c\u041e\u0429 \u041d\u0410 \u041f\u0420\u0415\u041f\u041e\u0414\u0410\u0412\u0410\u041d\u0415\u0422\u041e&nbsp;<\/strong><strong>\u041d\u0410 \u0423\u0415\u0411\u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418<br \/><\/strong><br \/><strong>\u0425\u0440\u0438\u0441\u0442\u043e \u0425\u0440\u0438\u0441\u0442\u043e\u0432, \u0425\u0440\u0438\u0441\u0442\u043e \u041a\u0440\u0443\u0448\u043a\u043e\u0432<br \/><\/strong><br \/><strong>Abstract<\/strong>. \u041f\u0440\u0435\u0437 \u0438\u0437\u043c\u0438\u043d\u0430\u043b\u0438\u0442\u0435 \u0442\u0440\u0438 \u0434\u0435\u0441\u0435\u0442\u0438\u043b\u0435\u0442\u0438\u044f \u0438 \u043f\u043e\u043b\u043e\u0432\u0438\u043d\u0430 \u0430\u0440\u0445\u0438\u0442\u0435\u043a\u0442\u0443\u0440\u0430\u0442\u0430 MVC&nbsp;\u0434\u043e\u043a\u0430\u0437\u0430 \u0441\u0432\u043e\u044f\u0442\u0430 \u043f\u043e\u043b\u0435\u0437\u043d\u043e\u0441\u0442 \u0432 \u0441\u043e\u0444\u0442\u0443\u0435\u0440\u043d\u0430\u0442\u0430 \u0438\u043d\u0434\u0443\u0441\u0442\u0440\u0438\u044f. \u0428\u0438\u0440\u043e\u043a\u0430\u0442\u0430 \u0443\u043f\u043e\u0442\u0440\u0435\u0431\u0430 \u043d\u0430 MVC \u0435&nbsp;\u043f\u0440\u0435\u0434\u043f\u043e\u0441\u0442\u0430\u0432\u043a\u0430 \u0437\u0430 \u043d\u0435\u0439\u043d\u043e\u0442\u043e \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435, \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u0438 \u0435\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e \u043f\u0440\u0438\u043b\u0430\u0433\u0430\u043d\u0435 \u0432 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0443\u0435\u0431\u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f.&nbsp;\u0412 \u0440\u0430\u0431\u043e\u0442\u0430\u0442\u0430 \u0441\u0435 \u0430\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u0442 \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442\u0438\u0442\u0435, \u043a\u043e\u0438\u0442\u043e \u0441\u0440\u0435\u0449\u0430\u0442 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u0442\u0435\u043b\u0438, \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438&nbsp;\u0438 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043f\u0440\u0438 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u0438 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0430\u0440\u0445\u0438\u0442\u0435\u043a\u0442\u0443\u0440\u0430\u0442\u0430 MVC. \u041f\u0440\u0435\u0434\u043b\u0430\u0433\u0430 \u0441\u0435&nbsp;\u043f\u043e\u0434\u0445\u043e\u0434 \u043d\u0430 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435, \u043a\u0430\u0442\u043e \u0441\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f\u0442 \u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430 \u0441\u043f\u0440\u0430\u0432\u044f\u043d\u0435 \u0441 \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442\u0438\u0442\u0435.&nbsp;\u041f\u043e\u0434\u0445\u043e\u0434\u044a\u0442 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u043f\u0440\u043e\u0446\u0435\u0441\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435, \u043a\u0430\u0442\u043e \u0440\u0430\u0437\u0433\u0440\u044a\u0449\u0430 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 MVC&nbsp;\u0432 \u043f\u043e\u0441\u043e\u043a\u0430 \u043e\u0442 \u0430\u0431\u0441\u0442\u0440\u0430\u043a\u0446\u0438\u044f\u0442\u0430 \u043d\u0430 \u043c\u043e\u0434\u0435\u043b\u0438 \u043a\u044a\u043c \u0440\u0435\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f \u043d\u0430 \u0438\u0437\u0445\u043e\u0434\u0435\u043d \u043a\u043e\u0434, \u0430\u043a\u0446\u0435\u043d\u0442\u0438\u0440\u0430\u0439\u043a\u0438&nbsp;\u0432\u044a\u0440\u0445\u0443 \u0437\u0430\u0432\u044a\u0440\u0448\u0435\u043d\u043e\u0441\u0442\u0442\u0430 \u043d\u0430 \u0441\u043e\u0444\u0442\u0443\u0435\u0440\u043d\u0430\u0442\u0430 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430. \u041e\u043f\u0438\u0441\u0430\u043d\u0438 \u0441\u0430 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438\u0442\u0435 \u043e\u0442 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0442\u0435 \u043f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438 \u0435\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0438 \u0441 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u0438 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438.&nbsp;<\/p>\n<p><em>Keywords<\/em>: model-view-controller, teaching methodology, web development<\/p>\n<p>&nbsp;<\/p>\n<p><strong>MODEL-VIEW-CONTROLLER ARCHITECTURE&nbsp;<\/strong><strong>HELPING IN TEACHING OF WEB TECHNOLOGIES<br \/><\/strong><br \/><strong>Abstract<\/strong>. During the last three and a half decades the MVC architecture has proven&nbsp;its usefulness in the software industry. The wide usage of MVC makes a strong case for&nbsp;its teaching, learning and effective application in the development of web applications&nbsp;by pupils and students.&nbsp;In the paper the difficulties that professors, students and pupils face when teaching and&nbsp;learning about the MVC architecture are analysed. A new way of approaching teaching is&nbsp;suggested as well as solutions to cope with the difficulties. The new approach considers&nbsp;the process of education by further developing the teaching of MVC in a certain direction:&nbsp;from the abstraction of models to the realisation of source code by putting the accent on&nbsp;the completion of the software product.<\/p>\n<p><strong>Hristo Hristov<\/strong><br \/>PhD student<br \/>Department of Methodology of Mathematics, Informatics and IT Education<br \/>FMI, University of Plovdiv<\/p>\n<p><strong>Hristo Krushkov<\/strong><br \/>Associate professor, PhD<br \/>Department of Computer Science<br \/>FMI, University of Plovdiv<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong><a name=\"art08\">&nbsp;<\/a>\u041f\u042a\u0420\u0412\u0418\u042f\u0422 \u0411\u042a\u041b\u0413\u0410\u0420\u0421\u041a\u0418 \u0423\u0427\u0415\u0411\u041d\u0418\u041a&nbsp;\u041f\u041e \u0422\u0415\u041e\u0420\u0418\u042f \u041d\u0410 \u0412\u0415\u0420\u041e\u042f\u0422\u041d\u041e\u0421\u0422\u0418\u0422\u0415<\/strong><\/p>\n<p><strong>\u041f\u043b\u0430\u043c\u0435\u043d \u041c\u0430\u0442\u0435\u0435\u0432<br \/><\/strong><br \/><strong>Abstract<\/strong>. \u0426\u0435\u043b\u0442\u0430 \u043d\u0430 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u0434\u0430 \u0437\u0430\u043f\u043e\u0437\u043d\u0430\u0435 \u0447\u0438\u0442\u0430\u0442\u0435\u043b\u044f \u0441 \u043f\u044a\u0440\u0432\u0438\u044f \u0443\u0447\u0435\u0431\u043d\u0438\u043a \u043f\u043e \u0442\u0435\u043e\u0440\u0438\u044f&nbsp;\u043d\u0430 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438\u0442\u0435 \u0437\u0430 \u0441\u0440\u0435\u0434\u043d\u043e\u0442\u043e \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u043d\u0430 \u0431\u044a\u043b\u0433\u0430\u0440\u0441\u043a\u0438 \u0435\u0437\u0438\u043a, \u0438\u0437\u0434\u0430\u0434\u0435\u043d \u043f\u0440\u0435\u0437 1890 \u0433.&nbsp;\u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u043e \u0435 \u043d\u0430\u043a\u0440\u0430\u0442\u043a\u043e \u0441\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435\u0442\u043e \u043c\u0443, \u043a\u043e\u0435\u0442\u043e \u043c\u043e\u0436\u0435 \u0434\u0430 \u043f\u043e\u0441\u043b\u0443\u0436\u0438 \u0437\u0430 \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u0441\u044a\u0441&nbsp;\u0441\u0435\u0433\u0430\u0448\u043d\u043e\u0442\u043e \u0441\u044a\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043d\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u043a\u0430 \u0443 \u043d\u0430\u0441.<\/p>\n<p><em>Keywords<\/em>: statistical education, history of mathematics<\/p>\n<p>&nbsp;<\/p>\n<p><strong>THE FIRST BULGARIAN TEXTBOOK&nbsp;ON PROBABILITY THEORY<br \/><\/strong><br \/><strong>Abstract.<\/strong> The purpose of this article is to acquaint the reader with the first textbook&nbsp;on probability theory for the secondary school in Bulgarian released in 1890. Its content is&nbsp;presented in brief and it could be compared with the current state of stochastic education&nbsp;in our country.<\/p>\n<p><strong>Plamen Mateev<br \/><\/strong>\u0002Associate Professor, Ph.D.<br \/>Statistical Laboratory<br \/>Department of Probability, Operations research and Statistics<br \/>Faculty of Mathematics and Informatics<br \/>Sofia University \u201eSt. Kliment Ohridski\u201c<br \/>Laboratory for the digitization of scientific and cultural heritage<br \/>Section of Information Systems<br \/>Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>","protected":false},"excerpt":{"rendered":"<p>&nbsp;\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS\u041f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0430 \u043a \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u043e\u043c \u2013 \u0443\u0434\u043e\u0431\u043d\u044b\u0439 \u043f\u043e\u0432\u043e\u0434 \u043e\u0431\u0441\u0443\u0434\u0438\u0442\u044c \u0442\u043e\u043d\u043a\u043e\u0441\u0442\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u0439 [Preparation for Olympiads \u2013 a Convenient Cause to Discuss&nbsp;Refinements of Mathematical Reasonings] \/ \u0412\u044f\u0447\u0435\u0441\u043b\u0430\u0432 \u0422\u0435\u043f\u043b\u044f\u043a\u043e\u0432, Vjacheslav&nbsp;Tepljakov \/ \u041d\u0438\u043d\u0430 \u041f\u0430\u0442\u0440\u043e\u043d\u043e\u0432\u0430, Nina Patronova &#8211; \u0441\u0442\u0440. 305 &nbsp; &nbsp; \u0412\u044a\u0440\u0445\u0443 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u0441\u044a\u0441 \u0437\u0440\u0438\u0442\u0435\u043b\u043d\u0438 \u0443\u0432\u0440\u0435\u0436\u0434\u0430\u043d\u044f[On Mathematics Teaching to Students with Visual Impairments] \/ [&hellip;]<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[2882],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - 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